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General Information
Course: Mathematics for Business I
Professor: Sarai Vera Rodríguez
University: UCAM Universidad Católica San Antonio
Syllabus
Unit 1: Vector Spaces
Unit 2: Matrix: Matrix calculus
Unit 3: Linear equations system
Unit 4: Linear Applications
Unit 5: Real Matrix Diagonalization
Unit 6: Quadratic real forms
Assessment Overview
Theoretical Part: 80%
Exams: Two written exams
Part 1: 30% (Midterm)
Part 2: 50% (Final)
Practical Part: 20%
Individual assignments (one assignment per unit).
Deliverables must be a single PDF document, well-written and legible.
To pass, average grade of all assignments must be 5 or greater.
Grading Criteria
Students pass if their weighted average is >= 5 after passing all evaluative components.
In case of failing any part (>20% weight), it must be retaken in the same academic year.
Key Concepts in Vector Spaces
Vector Space: An algebraic structure consisting of a set where vector addition and scalar multiplication are possible.
Properties of Vector Addition: 8 Properties
Associativity
Identity element
Inverse element
Commutativity
Properties of Scalar Multiplication: 5. Associativity 6. Identity element 7. Distributivity across vector addition 8. Distributivity across scalar addition
Linear Combinations: A linear combination of vectors is formed by multiplying and adding them using scalars.
Matrix and Operations
Matrix Representation: A way to organize a system of equations for solving.
Types of Matrices: Row, column, square, diagonal, scalar, and identity matrices.
Matrix Operations: Addition and scalar multiplication defined by element-wise operations.
Determinants
Definition: The determinant of a square matrix provides important properties regarding the matrix.
Properties: 1. Non-null if and only if the matrix is regular. 2. Properties regarding linear combinations and order.
Systems of Linear Equations
Identified as consistent (has solutions) or inconsistent (no solutions).
Consistent Determined: Unique solution.
Consistent Undetermined: Infinite solutions.
Solving Techniques
Cramer’s Rule: For unique solutions using determinants for matrix representation of equations.
Gauss-Jordan Method: Transform system into a diagonal matrix to find solutions.
Rouché-Frobenius Theorem
The system is inconsistent if the rank of the coefficient matrix and augmented matrix differ.
A consistent system exists if their ranks are equal.